An extension of maximum principle with some applications

Let U subset of R-n (res. D subset of R-n) be an open (res. a compact) subset, and let L be an elliptic operator defined on C-2(U, R) (res. C-2(D, R) ). In the present paper, we are going to extend the maximum principle for the function f is an element of C-2(U, R) (res. f is an element of C-2(D, R) ) satisfying the equation Lf = epsilon, where epsilon is a real everywhere nonzero continuous function on U (res. D ). Finally, we obtain some applications in mathematics and physics.